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Disjunction And Existence Properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Definitions * The disjunction property is satisfied by a theory if, whenever a sentence ''A'' ∨ ''B'' is a theorem, then either ''A'' is a theorem, or ''B'' is a theorem. * The existence property or witness property is satisfied by a theory if, whenever a sentence is a theorem, where ''A''(''x'') has no other free variables, then there is some term ''t'' such that the theory proves . Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: * The numerical existence property (NEP) states that if the theory proves (\exists x \in \mathbb)\varphi(x), where ''φ'' has no other free variables, then the theory proves \varphi(\bar) for some n \in \mathbb\text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Peter J
Peter may refer to: People * List of people named Peter, a list of people and fictional characters with the given name * Peter (given name) ** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church * Peter (surname), a surname (including a list of people with the name) Culture * Peter (actor) (born 1952), stage name Shinnosuke Ikehata, a Japanese dancer and actor * ''Peter'' (1934 film), a film directed by Henry Koster * ''Peter'' (2021 film), a Marathi language film * "Peter" (''Fringe'' episode), an episode of the television series ''Fringe'' * ''Peter'' (novel), a 1908 book by Francis Hopkinson Smith * "Peter" (short story), an 1892 short story by Willa Cather * ''Peter'' (album), a 1972 album by Peter Yarrow * ''Peter'', a 1993 EP by Canadian band Eric's Trip * "Peter", 2024 song by Taylor Swift from '' The Tortured Poets Department: The Anthology'' Animals * Peter (Lord's cat), cat at Lord's Cricket Ground in London * Peter (chief mouse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistency, consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of Mathematical proof, proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. Employing a Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Intuitionistic Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Axiomatization Heyting arithmetic can be characterized just like the first-order theory of Peano arithmetic , except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle , do not hold. Note that to say does not hold exactly means that the excluded middle statement is not automatically provable for all propositions—indeed many such statements are still provable in and the negation of any such disjunction is inconsistent. is strictly stronger than in the sense that all -theorems are also -theorems. Heyting arithmetic comprises the axioms of Peano arithmetic and the intended model is the collection of natural numbers . The signature includes zero "0" and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Recursively Enumerable Set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm can run forever, and no information is returned. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Harvey Friedman (mathematician)
Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years, this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete". Biography Friedman is the brother of mathematician Sy Friedman. Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, at age 19, with a dissertation on ''Subsystems of Analysis''. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Visiting Scientist at IBM. He delivered the Tarski Lectures in 2007. In 1967, Friedman was listed in the ''Guinness Book of World Records'' for being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 that satisfies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Epimorphism
In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category ''C'' is a monomorphism in the dual category ''C''op). Many authors in abstract algebra and universal algebra define an epimorphism simply as an ''onto'' or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Projective Object
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object. Definition An object P in a category \mathcal is ''projective'' if for any epimorphism e:E\twoheadrightarrow X and morphism f:P\to X, there is a morphism \overline:P\to E such that e\circ \overline=f, i.e. the following diagram commutes: That is, every morphism P\to X factors through every epimorphism E\twoheadrightarrow X. If ''C'' is locally small, i.e., in particular \operatorname_C(P, X) is a set for any object ''X'' in ''C'', this definition is equivalent to the condition that the hom functor (also known as corepresentable functor) : \operatorname(P,-)\colon\mathcal\to\mathbf preserves epimorphisms. Projective objects in abelian categories If the category ''C'' is an abelian category such as, for example, the category o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Free Topos
Free may refer to: Concept * Freedom, the ability to act or change without constraint or restriction * Emancipate, attaining civil and political rights or equality * Free (''gratis''), free of charge * Gratis versus libre, the difference between the two common meanings of the adjective "free". Computing * Free (programming), a function that releases dynamically allocated memory for reuse * Free software, software usable and distributable with few restrictions and no payment *, an emoji in the Enclosed Alphanumeric Supplement block. Mathematics * Free object ** Free abelian group ** Free algebra ** Free group ** Free module ** Free semigroup * Free variable People * Free (surname) * Free (rapper) (born 1968), or Free Marie, American rapper and media personality * Free, a pseudonym for the activist and writer Abbie Hoffman * Free (active 2003–), American musician in the band FreeSol Arts and media Film and television * ''Free'' (film), a 2001 American dramedy * ''Fre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
